Model and parameter selection for optical metrology

ABSTRACT

A profile model for use in optical metrology of structures in a wafer is selected, the profile model having a set of geometric parameters associated with the dimensions of the structure. The set of geometric parameters is selected to a set of optimization parameters. The number of optimization parameters within the set of optimization parameters is less than the number of geometric parameters within the set of geometric parameters. A set of selected optimization parameters is selected from the set of optimization parameters. The parameters of the set of selected geometric parameters are used as parameters of the selected profile model. The selected profile model is tested against one or more termination criteria.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a Continuation of U.S. patent application Ser. No. 10/206,491, filed Jul. 25, 2002, issued on Feb. 12, 2008, as U.S. Pat. No. 7,330,279, which is incorporated herein by reference in its entirety for all purposes.

BACKGROUND OF INVENTION

1. Field of Invention

This application relates to integrated circuit (IC) metrology and more particularly to the selection of a structure model and parameters for optical metrology.

2. Related Art

With the current drive towards smaller geometries of IC devices, measurement of IC device features is increasingly difficult as the size of the features become smaller. One method of measuring the features involves the use of gratings or periodic structures formed in test areas of a wafer that are proximate to or within the devices or circuits on the wafer. Knowledge of the dimensions of the gratings or periodic structures is essential in order to determine if the dimensions of the structure are within acceptable ranges and if, for example, a particular fabrication process causes the sidewalls of the features to be tapered, vertical, T-topped, undercut, or have footings.

Measurement of the periodic structure features may be done with a scanning electron microscope (SEM) or similar device where the sample is cleaved and examined. The cross-section SEM method is typically slow, expensive, destructive, and typically only provides one measurement number seen from the top of the feature.

Another measurement technique uses scatterometry. In scatterometry, spectroscopic reflectometry and ellipsometry, multiple-angle-of-incidence (MAI) devices, and mixed design systems are typically used to shine light on the structure and measure the reflected light. Empirical scatterometry basically uses an approach where the diffraction signals are measured for known widths of features of a structure, the pair of diffraction signals and structure widths used to create a library. Even for a limited library of structure dimensions and associated diffraction signals, the empirical scatterometry technique for building a library is time consuming and expensive. As the resolution of the structure dimension increases, the size of the library increases while the time to create and use the library increases considerably.

In another measurement technique, instead of using a library of diffraction signals and profile data, regression is used to determine the profile data from the measured diffraction signal. In using regression, one or more optimization techniques may be used to determine the profile data from the measured diffraction signal.

The length of time needed to create a library or to make a regression result converge varies depending on the profile model used and the number of parameters used to represent the profile model in the diffraction signal calculations. Typically, the more complicated the profile model and the more parameters used, the more time and/or computing resources needed to retrieve the desired information from measurements.

SUMMARY OF INVENTION

In an exemplary embodiment, a profile model for use in optical metrology of structures in a wafer is selected, the profile model having a set of geometric parameters associated with the dimensions of the structure. The set of geometric parameters is selected to a set of optimization parameters. The number of optimization parameters within the set of optimization parameters is less than the number of geometric parameters within the set of geometric parameters. A set of selected optimization parameters is selected from the set of optimization parameters. The parameters of the set of selected geometric parameters are used as parameters of the selected profile model. The selected profile model is tested against one or more termination criteria.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is an architectural diagram illustrating the use of optical metrology to measure the diffraction signals off wafer periodic structures.

FIG. 2 is an exemplary flowchart of the overall process for model and parameter selection for optical metrology of integrated circuit structures.

FIG. 3 is an exemplary flowchart for processing characterization of the wafer structure.

FIG. 4 is an exemplary flowchart for converting characterization of the wafer structure into a model and associated parameters.

FIG. 5 is an exemplary flowchart for selecting parameters of the model based on one or more selection criteria.

FIG. 6A is an architectural diagram depicting a system for model and parameter selection in an exemplary embodiment.

FIG. 6B is an architectural diagram of a metrology model optimizer in an exemplary embodiment.

FIG. 6C is an architectural diagram of a metrology model optimizer integrated with a wafer fabrication cluster.

FIG. 7 is an architectural diagram depicting the use of optimization engines in an exemplary embodiment.

FIG. 8 is an exemplary architectural diagram of a geometric model of the profile of a wafer structure.

FIG. 9A is an exemplary geometric shape utilized for building a model of the profile of a wafer structure.

FIG. 9B is an exemplary combination of geometric shapes utilized for building a model of the profile of a wafer structure.

FIG. 9C is an exemplary composite structure using a combination of geometric shapes as a model of the profile of a wafer structure.

FIG. 10A is an exemplary diagram of a wafer structure model using a rectangle and one trapezoid.

FIG. 10B is an exemplary reflectance graph of two highly correlated parameters of a wafer structure model using a rectangle and one trapezoid.

FIG. 10C is a table illustrating goodness of fit (GOF) and the confidence interval of each parameter of the model using a rectangle and one trapezoid.

FIG. 11A is an exemplary diagram of a wafer structure model using a rectangle and two trapezoids.

FIG. 11B is an exemplary reflectance graph of two highly correlated parameters of a wafer structure model using a rectangle and two trapezoids.

FIG. 11C is a table illustrating the goodness of fit (GOF) and confidence interval of each parameter of the profile model using a rectangle and two trapezoids.

FIG. 12A is an exemplary table of correlation coefficients of parameters of a wafer structure profile model.

FIG. 12B is an exemplary reflectance difference graph of two parameters of a profile model that have complete correlation.

FIGS. 13A to 13D are exemplary profile models using from one to four trapezoids to model a wafer structure.

FIG. 13E is an exemplary graph of the cost function and GOF of simulated signal versus the measured signal as a function of the number of trapezoids used in the profile model.

FIG. 14 is an exemplary graph of the cost function and GOF of simulated signal versus the measured signal as a function of the number of parameters used in the profile model.

FIG. 15 is a model and parameter selection data store layout in an exemplary embodiment.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENT(S)

In order to facilitate the description of the present invention, either an ellipsometric or reflectometric optical metrology system is used to illustrate certain concepts and principles. Graphs of diffraction signals off wafer structures using an ellipsometer, e.g., cos(Δ) and tan(Ψ), will be utilized to describe certain exemplary embodiments while reflectometric reflectance graphs will be utilized to describe others. It is understood that the same concepts and principles equally apply to ellipsometric, reflectometric, or other optical metrology systems.

FIG. 1 is an architectural diagram illustrating the use of optical metrology to measure the diffraction signals off structures patterned on a wafer. The optical metrology system 40 includes a metrology beam source 41 projecting a beam 43 at the target structure 59 of a wafer 47. The metrology beam 43 is projected at an incidence angle θ_(i) towards the target structure 59 and diffracted at a diffraction angle θ_(d) with respect to normal n. The diffraction beam 49 is measured by a metrology beam receiver 51. The diffraction beam data 57 is transmitted to a profile application server 53. The profile application server 53 compares the measured diffraction beam data 57 against a library 54 of calculated diffraction beam data representing varying combinations of critical dimensions of the target structure and resolution of the critical dimensions. In one exemplary embodiment, the library instance in library 54 best matching the measured diffraction beam data 57 is selected. The profile and associated critical dimensions of the selected library instance may provide a two-dimensional or three-dimensional representation of the grating structure. The optical metrology system 40 may utilize a reflectometer, an ellipsometer, or other optical metrology device to measure the diffraction beam or signal. An optical metrology system is described in co-pending U.S. patent application Ser. No. 09/727,530 entitled “System and Method for Real-Time Library Generation of Grating Profiles” by Jakatdar, et al., filed on Nov. 28, 2000, and is incorporated in its entirety herein by reference.

FIG. 2 is an exemplary flowchart of the overall process for model and parameter selection for optical metrology of wafer structures. In step 310, one or more termination criteria for selection of a structure profile model are determined. A termination criterion is a yardstick against which the result of the selection process is measured. The termination criteria may include a cost function value, a Goodness-of-Fit (GOF) value, and/or other curve fitting metrics, as well as confidence intervals of parameters measured.

For example, a cost function between a simulated signal off the structure using the profile model parameters compared to a measured signal may be used as a termination criterion. One cost function comparison is illustrated by the equations below, where V₁ and V₂ are two vectors of size n, and the cost function of V₁ relative to V₂ is:

$\begin{matrix} {{{Cost}\left( {V_{1},V_{2}} \right)} = \left( {\sum\limits_{i = 1}^{n}\left( {V_{1i} - V_{2i}} \right)^{p}} \right)^{1/p}} & (1.00) \end{matrix}$ where i represents the i th member of the vector and p is an arbitrary number associated with the metric. The first vector is the set of signal values at measurement points for the metrology device used and the second vector is the corresponding set of simulated signal values at the same points. A cost function termination criterion may be set at a specific number, for example, 0.25. Another termination criterion may be the goodness of fit (GOF) between the graph of the measured and simulated signal values. The GOF is a measure of the proximity of two sets of values. For example, when ellipsometric measurements are used, GOF is based on values for tan Ψ and cos Δ, where tan Ψ and cos Δ are represented by a single vector of n dimensions: S=[tan Ψ₁ tan Ψ₂ . . . tan Ψ_(n/2) cos Δ₁ cos Δ₂ . . . cos Δ_(n)]  (1.10)

One commonly used formula for GOF between a measured signal S_(m) compared to a simulated signal S_(s) is:

$\begin{matrix} {{G\; O\; F} = {1 - \frac{\sum\limits_{i}^{n}\left( {{S_{s}(i)} - {S_{m}(i)}} \right)^{2}}{\sum\limits_{i}^{n}\left( {{S_{m}(i)} - {\overset{\_}{S}}_{m}} \right)^{2}}}} & (2.00) \\ {{{where}\mspace{14mu}{\overset{\_}{S}}_{m}} = \frac{\sum\limits_{i}^{n}{S_{m}(i)}}{n}} & (2.10) \end{matrix}$ where i represents the i th measurement point for an optical metrology device, n is the total number of measurements for the optical metrology device.

Another termination criterion is a confidence interval cutoff value for optimization parameters. Optimization parameters and confidence intervals are explained in more detail below. Associated with a profile model is a set of geometric parameters. Optimization parameters are derived from the geometric parameters of the profile model. The process of deriving the optimization parameters from the geometric parameters will also be discussed in detail in FIG. 4. Confidence interval is a range of values of the optimization parameter within which the actual value is expected to fall with a specified probability. As an illustration, a 3-sigma confidence interval of an optimization parameter x1 of 20 nm means there is a 99.7% probability that the actual value of x1 is within+ or −20 nm. The confidence interval amount may be set to the amount of change from the nominal value of an optimization parameter where the change in the diffraction signals is greater than a preset value. The preset value may be a value for system noise level or artificial noise level, typically expressed in nanometers. For example, a confidence interval cutoff of 2.0 nm for the middle CD of a structure and 2.5 nm for the bottom CD parameter may be specified. The selection of the profile model would continue until the confidence interval cutoff for both the middle and bottom CD's are met.

In step 320, one or more criteria for selection of profile model parameters are determined. Profile selection criteria may include a specific correlation coefficient or sensitivity of a profile parameter. For example, a cutoff of 0.99 correlation may be used to select parameters. Alternatively, a specific change in the signal (ΔS) may be required for each incremental change of the profile parameter (ΔP), where ΔS/ΔP is a measure of the sensitivity of the parameter. The parameter selection criteria will be discussed in more detail in FIG. 5.

Still referring to FIG. 2, in step 330, the characterization of the wafer structure is obtained. An image of the structure from a referencing metrology instrument, such as a cross section-SEM or X-SEM image may be used as a basis for characterizing the profile of the structure. For example, indication of top-rounding, undercutting, T-topping, footing, notching, concavity, convexity, and similar characterization of the structure may be obtained. Wafer fabrication process design data may be used. Information about the nominal CD and height together with structure image data may be used to characterize the structure profile. For a description of the steps involved in one exemplary process of obtaining characterization of the wafer structure, refer to the description of FIG. 3.

In step 340, the structure characterization is converted into a profile model. A profile model may be a simple rectangle with two parameters designating the height and width of the rectangle. In FIG. 9A, a rectangular shape 920 is shown with two parameters, a0 and a1 representing the width and height respectively. Another model for a profile may be a trapezoid, having three parameters representing for instance the bottom CD, the top CD, and the height. FIG. 9B represents a model with two trapezoids 925, one on top of the other. The two-trapezoid profile model 925 could be described using five parameters, a0 representing the top CD of the top trapezoid, a1 representing the common CD of the top and bottom trapezoids, a2 representing the bottom CD of the bottom trapezoid, a3 representing the total thickness of the top and bottom trapezoids, and a4 representing the thickness of the top trapezoid. A more complex profile model is shown in FIG. 9C with two rectangular blocks, 942 and 944, representing two different films; a trapezoid 940 that represents the bottom of the patterned structure having a footing, two rectangular blocks, 938 and 936, made of a different material, a thin rectangular block 934 representing a notch in the structure, and top trapezoid 932 representing some level of rounding in the top of the structure.

As can be seen in FIG. 9C, a profile model may comprise many different geometric shapes in order to get a good approximation of the actual profile of the structure. In general, the more complex the model, the more parameters needed to represent the model. More parameters increase the complexity and length of time to perform the optical metrology simulation of the structure. For a description of simulation of diffraction signals off a hypothetical structure, refer to co-pending U.S. patent application Ser. No. 09/770,997, entitled “Caching of Intra-layer Calculations for Rapid Rigorous Couple-Wave Analyses”, by Niu et al., filed on Jan. 26, 2000, incorporated in its entirety herein by reference. As will be described in FIG. 5, for a given profile model, the number of parameters may be optimized in order to select the least number of parameters that still meet the termination criteria.

In step 350 of FIG. 2, the profile model optimization parameters are selected based on one or more selection criteria. As will be discussed in more detail in FIG. 5, the selection of an optimization parameter is based on correlation with other parameters, sensitivity of the simulated signal to a change of the optimization parameter, confidence interval of parameter change that can be detected, and other considerations. Stated another way, and as an example of an exclusion rule, an optimization parameter A may be excluded if parameter A is highly correlated to another parameter B and the simulated signal is insensitive to changes in parameter A.

In step 360, the simulation calculation is optimized by balancing the speed of simulation computations with the accuracy of the computed signal. For example, variables such as number or range of diffraction wavelengths used and the number of diffraction orders considered are optimized to yield the least number of simulation variables and the highest accuracy of the computed signal.

In step 370, a test is performed to see if the termination criteria are met. For example, if one of the termination criteria is a cost function value of less than or equal to 2.50, then the cost function value of a simulated signal using the selected parameters of the selected model is compared to a corresponding measured signal. If the cost function value is 2.20, then this criterion is met. Additionally, a second termination criterion may be a GOF of 0.9990 or greater. Referring to FIG. 11B, the graph 982 of reflectance on the Y-axis as a function of wavelength on the X-axis, measured reflectance curve 984 is compared to simulated reflectance curve 986, the simulation using a double trapezoid profile as illustrated in FIG. 11A. Using the GOF formula, the calculated GOF is 0.9994 as shown in FIG. 11C. However, in the same table 988 in FIG. 11C, the highest 3-sigma confidence interval for optimization parameters is 17.92 nm for x1. As noted above, the confidence interval is a range of values of the optimization parameter within which the actual value is expected to fall with a specified probability. As an illustration, a 3-sigma confidence interval of an optimization parameter x1 of 20 nm means there is a 99.7% probability that the actual value of x1 is within+ or −20 nm.

Referring now to FIG. 10B, the graph 960 of reflectance on the Y-axis as a function of wavelength on the X-axis, measured reflectance curve 962 is compared to simulated reflectance curve 964, the simulation using a single trapezoid profile as illustrated in FIG. 10A. Using the GOF formula, the calculated GOF is 0.9990 as shown in FIG. 10C. Note that in table 965 of FIG. 10C, the highest 3-sigma confidence interval for the optimization parameters is 1.99 nm for x0. As explained in more detail below, the lower 3-sigma confidence interval of the single trapezoid model of FIG. 10A compared to the two trapezoid model of FIG. 11A means that the single trapezoid model of FIG. 10A will be selected, given that the GOF criterion of 0.9990 is also met.

Referring now to FIG. 2, when the termination criteria are not met, processing proceeds to step 375, where the parameter selection criteria and/or the profile model is adjusted, and steps 350, 360, and 370 are iterated. Examples of changes to parameter selection criteria may be an adjustment of the correlation cutoff for selecting or excluding a parameter. Alternatively, a sensitivity cutoff, expressed as sum-squared-error values as an example, may be adjusted. An example of a profile model adjustment is using three trapezoids instead of two trapezoids to represent the structure profile or using one trapezoid instead of two trapezoids to model the patterned area of the structure. In one instance, the profile model may be revised to include more or different geometric shapes to get closer to the optical microscopy image of the structure. In another instance, the profile model may be made simpler, such as using only one trapezoid instead of several trapezoids.

In step 380, when the termination criteria are met, the profile model, the selected profile parameters, the parameter selection criteria, the termination criteria, and identification data regarding the fabrication, wafer site, and metrology device are saved in a data store.

The results of model and parameter selection may be utilized in several ways. In step 390, a library of simulated diffraction signals and associated profile data is created using the ranges and resolutions of the selected parameters of the selected model. For a description of the process for creating a library using ranges and resolutions of parameters, refer to co-pending U.S. patent application Ser. No. 09/727,530 entitled “System and Method for Real-Time Library Generation of Grating Profiles” by Jakatdar, et al., filed on Nov. 28, 2000, and is incorporated in its entirety herein by reference. Alternatively, in step 395, the results of model and parameter selections are displayed. In one embodiment, the values of the critical dimensions, profile shape, and film thickness are made available as soon as the one or more termination criteria are met. In another embodiment, some or all of the data saved in step 390 are displayed. In still another embodiment, in step 398, the results of profile model and parameter selection are utilized for fabrication cluster feed-forward or feed-backward control loops. Details of this aspect are discussed in FIG. 6C.

As noted above, the description of FIG. 3 provides more detail regarding the overall flowchart step of obtaining and processing characterization of the wafer structure. It is understood that the process described in the following steps is but one technique of obtaining the characterization of the wafer structure. Other techniques may include structure characterization obtained from an integrated circuit fabrication process or from integrated circuit device simulation software.

Referring to FIG. 3, in step 410, one or more data gathering criteria is set. A data-gathering criterion is used to test whether sufficient data about the structure is available to perform the model and parameter selection. Examples of data gathering criterion may be a cost function value or GOF similar to the termination criteria used in model and parameter selection. However, the cost function value or GOF may be different, typically lower, from those specified for the termination criteria. Other data gathering criteria may include a range of acceptable variation of measured diffraction signals, such as 3-sigma width of measured diffraction signals for the same site in the wafer. For example, if the measured diffraction signals for the same site in a wafer have large standard deviation, then additional diffraction signals measurements of the wafer structure may be needed. Alternatively, the data gathering criteria may be a comparison of the structure profile derived from simulations to an X-SEM image.

Still referring to FIG. 3, in step 420, characterization about the layer stack, unpatterned layer thickness, index of refraction index n, extinction coefficient k, and other layer properties are obtained. Characterization includes the type of material used in each layer. In step 430, the pitch of the patterned structure, line-to-space ratio, optical characteristics of the patterned structure, and other characterization of the patterned structure profile are obtained. Other characterization of the patterned structure profile includes data about top rounding, undercut, footing, notching, or other expected anomalies in the profile.

In step 440, measured optical metrology diffraction signals are selected from the input measured diffraction signals. The type and amount of data varies according to whether an ellipsometer, reflectometer, or other scatterometric device is used, and depending on the manufacturer of the device. Selection of measured diffraction signals involves several steps designed to test a small number of representative diffraction signals using selection techniques such as clustering, correlation, and the like. The measured diffraction signals are categorized into groups using one or more of the selection techniques listed above. For a description of clustering in optical metrology, refer to co-pending U.S. patent application Ser. No. 09/737,705 entitled “System and Method for Grating Profile Classification” by Doddi, et al., filed on Dec. 14, 2000, incorporated in its entirety herein by reference. Representatives of each cluster or group of highly correlated measured diffraction signal are identified and selected for use in the model and parameter selection processing.

In step 450, the signal off a structure is simulated utilizing the layer stack and structure profile developed from the characterization of the profile. For a description of simulation of diffraction signals off a hypothetical structure, refer to co-pending U.S. patent application Ser. No. 09/770,997, entitled “Caching of Intra-layer Calculations for Rapid Rigorous Couple-Wave Analyses”, by Niu et al., filed on Jan. 26, 2000, incorporated in its entirety herein by reference.

In step 460 of FIG. 3, a test is performed to see if the one or more data gathering criteria are met. For example, if the GOF between the simulated signal and the measured diffraction is 0.950 and a data gathering criterion is a GOF of 0.950 or lower, then the data gathering criterion is met. In another example, the data-gathering criterion is a simulated thickness of each layer of the stack being the same or within a given percent of the characterization data provided by the user. Assume the thickness of layer of the stack is given as 100 nm and the simulated thickness for that layer is 102 nm, and assuming one data gathering criterion is a variance of 2 percent or less on layer thickness, then the data gathering criterion is met.

Still referring to FIG. 3, in step 480, the data gathering criteria, wafer and structure characterization, and metrology device identification data are saved. If the data gathering criteria are not met, in step 470, additional characterization data is obtained or the data gathering criteria are adjusted. For example, if a data-gathering criterion is a cost function value of the simulated signal and measured signal of 3.50 or better, and the computed cost function value is 7.00, then the data gathering criterion is not met. A basic characterization data may be off. For example, if the pitch of the structure is specified incorrectly or the profile characterization is grossly incorrect, the cost function value could be very high. A review of the characterization data and accuracy of input of these characterizations into the system may be used to identify the cause of the problem. Alternatively, the data gathering criteria may be adjusted if found to be set incorrectly.

As noted above, the description of FIG. 4 that follows provides more detail regarding the overall flowchart step of converting characterization of the wafer structure into a model and associated parameters. In step 510, the types of geometric shapes for each material of the stack are determined. For example, where there is only one material in a stack, one geometric shape may be chosen to represent the entire profile model. In FIG. 8, assuming structure 900 is formed of one material, a set of rectangular shapes of varying dimensions (e.g., rectangular shapes 902, 904, 906, 908, and 910) is used to represent the profile model of structure 900. In FIG. 9A, for an unpatterned film, a rectangular shape is used, whereas in FIG. 9B, two trapezoidal shapes are used. The profile in FIG. 9C utilizes rectangular and trapezoidal shapes.

With reference to FIG. 4, in step 520, the geometric shapes and parameters of the stack of the structure are generated. For example, if the first layer in an unpatterned stack is represented by the rectangular shape 920 in FIG. 9A, then the geometric parameter is the thickness of the first layer, a1, since the width for an unpatterned layer can be assumed to be infinite for diffraction simulation purposes.

If the geometric shape for a layer is a trapezoid, three geometric parameters can be used, namely, the top width, the bottom width, and height of the trapezoid. If a double-trapezoid 925 is used as in FIG. 9B, then five geometric parameters can be used, namely, the top width of the top trapezoid a0, the bottom width of the top trapezoid a1, which is also the top width of the bottom trapezoid, the bottom width of the bottom trapezoid a2, the total thickness of the structure model a3, and the thickness of the top trapezoid a4.

The profile model 930 of FIG. 9C depicts a complex profile model where the model includes rectangular shapes 936, 938, 942, and 944, a rectangular shape to illustrate notching in the structure 934, and trapezoidal shapes 932 and 940 to illustrate a top rounding of the structure and a bottom footing of the structure respectively. The geometric parameters are the sum of individual geometric parameters of the individual geometric shapes. For the complex profile model 930 of FIG. 9C, the number of geometric parameters is high. Typically, the more geometric shapes, the higher the number of geometric parameters. Furthermore, the higher number of geometric parameters of a profile model results in a longer simulation process for determining the simulated diffraction signals. As mentioned previously, a longer diffraction simulation process may result in a considerably longer library creation time or regression time.

In step 530 of FIG. 4, the nominal values and ranges of the geometric parameters are obtained. These values and ranges are typically obtained from historical or test data for the fabrication process or recipe. For example, a top width or top CD may have a nominal value of 200 nm and a range of 120 to 280 nm.

In step 540, the dependencies of the geometric parameters are defined. Again, the dependencies of the geometric parameters are based on historical or test results for the particular fabrication process of recipe. For example, in a shallow trench isolation (STI) structure having a silicon nitride cap and a silicon trench, the nitride cap typically determines the CD of the top width of the silicon trench. In this case, the independent geometric parameter is the nitride cap bottom CD. The top CD of the nitride cap and the top width of the silicon trench may be tied to the bottom CD of the nitride cap.

Using the double-trapezoid model 925 of FIG. 9B as an example, the top width a0 of the top trapezoid may be a function of the bottom width a 1 of the top trapezoid; a0 may have a linear relation to a1; for example, a 0 may be equal to a1 plus a constant or a0 may be equal to a1 multiplied by a fixed number. The relation of a geometric parameter to another geometric parameter may be characterized by a simple linear function, a quadratic function, polynomial function or the like. Dependencies of the geometric parameters of the profile model are defined based on whether a geometric parameter is an independent parameter, has a fixed offset from other parameters, has a variable offset from other parameters, or has a fixed value. For the sake of illustration, consider the double trapezoid of FIG. 9B having five geometric parameters. From design or previous experience with the fabrication recipe, a0 ay be known as an independent parameter. Also from previous fabrication data, a1 may be known to have an constant offset from a0 of 10 nm, a2 has a variable offset from a0, a3 is a constant, and a4 is two times a0.

Still referring to FIG. 4, in step 550, the geometric parameters are converted to optimization parameters, x. Reasons for conversion of geometric parameters into optimization parameters include reduction of the search space for regression to determine the optimized simulation diffraction signal (discussed later in FIG. 5). Another reason for conversion of geometric parameters into optimization parameter is reduction of correlation of a parameter to the other parameters.

The result of the conversion is an equation in terms of the optimization parameter x_(i). For example, the equation for each geometric parameter a_(i of) the double trapezoid shown in FIG. 9B, having the dependencies described above is as follows: a0=x0, a1=x0+10, a2=x0+x1, a3=50 a4=2x0, where a0, a1, a2, a3, and a4 are the geometric parameters of the profile model as defined above, expressed in nanometers, and x0 and x1 are the optimization parameters of the profile model. Note that the five geometric parameters have been converted into two optimization parameters. It should be noted that more complicated profile models may typically require more geometric parameters and may generally require a corresponding higher number of optimization parameters. It is understood to a person knowledgeable in the art, that other equivalent ways of expressing the dependencies of the geometric parameters to optimization parameters may be used.

As noted above, the description of FIG. 5 that follows provides more detail regarding the overall flowchart step of selecting parameters of the model based on one or more selection criteria. Referring to FIG. 5, in step 810, the optical metrology wavelengths or range of wavelengths for profile model selection are selected. For a description of the process to select wavelengths, refer to co-pending U.S. patent application Ser. No. 10/162,516, entitled “Selection of Wavelengths for Integrated Circuit Optical Metrology”, by Doddi, et al., filed on Jun. 3, 2002, incorporated herein in its entirety by reference. Several tasks may be concurrently or serially performed to provide information as to whether an optimization parameter should be selected or excluded.

In step 820, the correlation between the optimization parameters is determined. Typically, a correlation coefficient, r, between two optimization parameters is calculated using the formula:

$\begin{matrix} {r = \frac{\sum\limits_{i}{\left( {x_{i} - \overset{\_}{x}} \right)\left( {y_{i} - \overset{\_}{y}} \right)}}{\sqrt{\sum\limits_{i}\left( {x_{i} - \overset{\_}{x}} \right)^{2}}\sqrt{\sum\limits_{i}\left( {y_{i} - \overset{\_}{y}} \right)^{2}}}} & (2.60) \end{matrix}$ where x_(i) and y_(i) is a pair of optimization parameters, x is the mean of x_(i)'s and y is the mean of y_(i)'s. The value of r lies between −1 and +1 inclusive. A correlation coefficient value of +1 can correspond to complete positive correlation and a value of −1 can correspond to complete negative correlation. A value of r close to zero can correspond to the x and y optimization parameters not being correlated.

Referring to FIG. 12A, the table of correlation coefficients 996 shows five optimization parameters, namely, x0 representing the resist top CD, x1 representing the resist bottom CD, x2 representing the resist thickness, x3 representing the anti-reflective coating thickness, and x4 representing the silicon dioxide thickness. The correlation coefficient table 996 is configured such that the Y-axis and the X-axis have the parameter numbers shown. An intersection or cell represents the correlation coefficient of a parameter matched to a different parameter. For example, at the intersection of parameter x0 and x4, cell 997, the correlation coefficient is 0.47. The correlation coefficient is calculated by substituting input measured values of x0 for x_(i) and x4 for y_(i) in equation 2.60 above. The low correlation coefficient value means that parameter x0 and x4 are not highly correlated. In contrast, at the intersection of parameter x3 and x4, cell 998, the correlation coefficient is 1.00, meaning complete positive correlation between x3 and x4. As can be seen in FIG. 12B, the reflectance difference graph versus wavelength of parameter x3, graph 992, has complete positive correlation to the reflectance difference graph versus wavelength of parameter x4, graph 994. In terms of optimization parameter selection, only one of parameter x3 and x4 need to be included, since the variation of the diffraction signals caused by either x3 or x4 can be determined from the variation of the other.

In step 830 of FIG. 5, the confidence interval of each optimization parameter is determined. As previously noted, the confidence interval may be set to the amount of change from the nominal value of an optimization parameter where the change in the diffraction signals is greater than the noise level. The noise in the diffraction signals may be due to system noise, for example, noise from the measurement devices, or the noise may be simulated. The confidence interval is generally expressed as a multiple of the standard deviation sigma, σ, of the optimization parameter. The standard deviation for an optimization parameter is calculated from measured values of the optimization parameter, using the formula: σ=√{square root over ((([1/(N−1)])*(x _(i) −x _(av))²))}{square root over ((([1/(N−1)])*(x _(i) −x _(av))²))}  (2.70) where N is the number of measurements, x_(i) is the i th value of the optimization parameter x, and x_(av) is the average value of the optimization parameter x.

The confidence interval is typically calculated from a given set of sample input data representing actual measurements off the wafer structure. The confidence interval may also be calculated using simulated random noise introduced in the measurement data for the optimization parameter.

With reference to FIG. 10A, the structure profile model 950 using a single trapezoid 951 on top of a rectangular shape 953 representing a structure with a single layer of underlying film has four optimization parameters, namely, x0 representing the top CD of the structure, x1 representing the bottom CD of the structure, x2 representing the width of the structure, and x3 representing the width of the underlying film. With reference to FIG. 10C, a table 965 is shown with the 3-sigma confidence interval for the four-optimization parameters. For example, optimization parameter x0 has a confidence interval of 1.99 nm, meaning that measurement of x0 has a probability of 99.7% being visible or sensitive to within 1.99 nm. Alternatively, if the change in x0 is less than 1.99 nm, then there is a 99.7% probability the change would not show in the signal. Similarly, x1 has a 3-sigma confidence interval of 1.95 nm, and so on. Note that parameter x3 has a 3-sigma confidence interval of 0.30, meaning that parameter x3 is sensitive to changes greater than 0.30 nm.

Assuming that the same structure as discussed in FIG. 10A was modeled using two trapezoids 972 on top of a rectangular shape 974 as in the profile model 970 in FIG. 11A. The double-trapezoid profile model 970 has six optimization parameters, namely, x0 representing the top CD of the top trapezoid 976, x1 representing the middle CD of the bottom trapezoid 978, x2 representing the bottom CD of the bottom trapezoid 978, x3 representing the width of the double trapezoid 972, x4 representing the ratio at the inflection point, equal to h1, the width of the top trapezoid 976, over the width of the double trapezoid 972, and x5 representing the width of the underlying film. With reference to FIG. 11C, a table 988 is shown with the 3-sigma confidence interval for the six-optimization parameters. For example, optimization parameter x0 has a confidence interval of 6.81 nm, meaning that measurement of x0 has a probability of 99.7% being visible or sensitive to within 6.81 nm. Alternatively, if the change in x0 is less than 6.81 nm, there is 99.7% probability the change in x0 would not show in the signal. As will be discussed in a later step 870, the entire collection of data calculated for each optimization parameter can be integrated into a decision-making step as to whether to include or exclude an optimization parameter.

In step 840 of FIG. 5, the sensitivity of the simulated signal to changes in one or more optimization parameters is determined. Typically, this determination is done by changing one optimization parameter by a small amount and keeping the other optimization parameters constant. For example, in the profile model in FIG. 10A using one trapezoid, the sensitivity of parameter x0 may be tested by adding one nanometer to the nominal value while keeping x1, x2, and x3 at nominal value and simulating the signal. If there is no noticeable change in the signal matrix or graph of (x0 at nominal plus 1 nm), then x0 has low sensitivity. The other optimization parameters can similarly be changed while holding the rest constant in order to test the sensitivity of each optimization parameter.

The sensitivity of an optimization parameter may be quantitatively expressed by calculating the sum-square-error (SSE) of the changed signal compared to the signal using nominal values. The SSE formula is as follows:

$\begin{matrix} {{S\; S\; E} = {\sum\limits_{i = 1}^{n}\left( {{S_{o}(i)} - {S_{1}(i)}} \right)^{2}}} & (3.00) \end{matrix}$ where i is the signal simulation point, typically at a preset wavelength, n is the number of signal simulation points, S₀ is the simulated signal value using nominal values of optimization parameters, S₁ is the simulated signal value using nominal plus change in one of the optimization parameters.

In step 870 of FIG. 5, the final selection of optimization parameters is made based on one or more selection criteria. For example, a parameter selection criterion may be a cutoff point in the correlation coefficient. Parameters with a correlation coefficient lower than 0.50 with respect to all other parameters may be selected. Alternatively, a pair of parameters with a correlation coefficient of 0.98 may be further tested as to sensitivity in order to determine which parameter is selected, or which parameter is excluded. An SSE threshold may be used to select optimization parameters. For example, an SSE threshold of 0.01 may be used to filter optimization parameters that are relatively insensitive to changes of the parameter. A certain cutoff for the 3-sigma confidence interval may also be used to screen out optimization parameters that do not meet these criteria or to flag profile models that do not yield the proper sensitivity of key parameters critical to the IC design. A combination of the above criteria may be used. It is understood that other equivalent criteria known to one knowledgeable in the art may be used. If an optimization parameter is not selected, then the optimization parameter is set to a fixed value, the fixed value determined from fabrication data or previous experience with the recipe.

In step 890, a procedure is performed to determine the optimized simulation diffraction signal to the measured signal using the selected optimization parameters of the selected profile model. One embodiment uses a regression technique to get to the optimized simulation signal. One or more types of regression engines may be used.

With reference to FIG. 7, a profile model tester 1400 receives the selected profile model, selected optimization parameters, and measured diffraction signals 1402. The profile model tester 1400 processes the input data and activates one or more optimization engines, such as branch-and-bound technique 1420, simulated annealing 1430, genetic algorithm 1440, other global optimization technique 1450, or hybrid global and local optimization technique 1460. The optimization engines arrive at a global minimum of the difference between the simulated signal and the measured signal. The simulated signal corresponding to the global minimum in turn corresponds to a set of values of the optimization parameters of the selected profile model, which the profile model tester 1400 creates as an output 1404.

FIG. 6A is an architectural diagram depicting a system for model and parameter selection in an exemplary embodiment. In the present embodiment, a terminal 1800 is used to enter wafer fabrication process design data including the stack, n and k values, nominal profile parameter values and ranges, width nominal values and ranges, measured diffracted signals off several sites in the wafer, and structure image data to characterize the structure profile. Choices of termination and optimization parameter selection criteria may be entered on the terminal 1800 and transmitted as input 1801 to a profile compiler 1810 and input 1803 to a model generator 1820. The profile compiler 1810 edits the input data 1801 and invokes an optical metrology simulator 1860 to simulate the signal with the specified nominal values of the geometric parameters of the profile model 2000. The optical metrology simulator 1860 transmits the simulated diffraction signal 2001 to the profile compiler 1810. As mentioned above, a description of simulation of diffraction signals off a hypothetical structure, refer to co-pending U.S. patent application Ser. No. 09/770,997, entitled “Caching of Intra-layer Calculations for Rapid Rigorous Couple-Wave Analyses”, by Niu et al., filed on Jan. 26, 2000, incorporated in its entirety herein by reference. The profile compiler 1810 performs a comparison of the simulated signal 2001 to the measured signal from the input 1801, sending data to terminal 1800 regarding the quality and adequacy of the input data 1801. The profile compiler 1810 may also process adjusted profile model data 2012 from a profile model tester 1840.

Referring to FIG. 6A, the profile compiler 1810 transmits the edited characterization data and measured diffraction signals 1811 to a model generator 1820. The model generator 1820 creates a profile model of the structure comprising geometric shapes. The geometric shapes are expressed in terms of geometric parameters and converted into optimization parameters 1821, which are transmitted to a parameter selector 1830. The parameter selector 1830 uses optimization parameter selection criteria to select which optimization parameters meet the correlation coefficient cutoff, the sensitivity threshold, and/or confidence interval requirements from the customer. The parameter selector 1830 invokes the optical metrology simulator 1860 to perform simulations of diffraction signals with profile parameter data 2004. In turn, the optical metrology simulator 1860 performs simulation of the diffraction signal and transmits simulated diffraction signals 2005 to the parameter selector 1830.

Part of the parameter selector 1830 function is to perform a procedure to determine the optimized simulated signal for each measured signal, invoking one or more optimization engines discussed in FIG. 7. After the optimization process, the optimized profile data comprising profile, CD, and film thickness is transmitted as output 2006 to critical dimension server 1870. The parameter selector 1830 transmits the selected optimization parameters 1831 to the profile model tester 1840, where the termination criteria such as cost function value, GOF, and/or other termination criterion are tested. If the termination criteria are not met, the profile model tester 1840 adjusts the profile model, for example, by switching from a two-trapezoidal model to a single trapezoidal model or switching from a simple geometric model to one using more geometric shapes to approximate the profile model more closely. The adjusted profile model 2012 is transmitted to the profile compiler 1810. Alternatively, if the termination criteria are met, the profile model tester 1840 stores the profile model termination criteria, optimization parameter selection criteria, fabrication process, wafer site, optical metrology device identification data, and selected optimization parameters 2010 in a data store 1850. The profile model tester 1840 transmits the optimization parameters 1841 to a library generator 1880, which creates a library 1890 comprising diffraction signals and associated profile data 1881, using the ranges and resolution of the selected optimization parameters.

FIG. 6B is an architectural diagram depicting a system for model and parameter selection in an exemplary embodiment. Metrology model optimizer 1900 receives requests 1902 for critical dimensions, profiles, and film thickness of measured diffraction signals from a profiler workstation 1910. Based on this request 1902 and other input data (not shown) characterizing the subject structure on the wafer, metrology model optimizer 1900 selects a model and parameters in a process similar to that described in FIG. 6A. The metrology model optimizer 1900 produces the requested critical dimensions, profiles, and film thickness associated with a measured diffraction signals and transmits these results 1901 back to the profiler workstation 1910. The profiler workstation 1910 may be located at a remote user site. Access to the metrology model optimizer 1900 may be through a private network or a public network like the Internet.

FIG. 6C is an architectural diagram of a metrology model optimizer in an exemplary embodiment. The system configuration is similar to the system in FIG. 6B except that instead of processing request for critical dimension data 1924 exclusively from the profiler workstation 1925, in-line requests 1931 for the same data is transmitted from an optical metrology system 1930. The optical metrology system 1930 is coupled to a fabrication cluster 1940, which may be a clean track unit, lithography machine, an etch machine or a combined lithography-and-etch unit. As a wafer (not shown) completes a fabrication process step, structures on the wafer are measured by the optical metrology system 1930 creating measured diffraction signals 1931 transmitted to the metrology model optimizer 1920. In addition to the critical dimension data 1924 being transmitted to the profiler workstation 1925, the same data is transmitted to the fabrication cluster 1940 for advanced process control use. The critical dimension data 1924 may be used by the fabrication cluster 1940 to adjust process variables of the fabrication process. The profiler workstation 1925 sends requests 1926 for critical dimensions, profiles, and film thickness of measured diffraction signals and other input data (not shown) characterizing the structures on the wafer or location of similar data stored in the metrology model optimizer 1920. The optical metrology system 1930 receives transmitted data 1941 from the fabrication cluster 1940 regarding completion of one or more fabrication processes. After completing the measurements of structures on the wafer, the optical metrology system transmits signals 1941 to the fabrication center 1940 to indicate completion of optical metrology measurements.

FIGS. 13A, 13B, 13C, and 13D are exemplary structure profiles using different profile models. FIG. 13A illustrates a structure modeled with a single trapezoid T1, whereas FIG. 13B illustrates the same structure modeled with two trapezoids T1 and T2. In similar manner, FIG. 13C illustrates the same structure modeled with three trapezoids T1, T2, and T3, whereas FIG. 13D illustrates the same structure modeled with four trapezoids T1, T2, T3, and T4.

As can be seen in FIGS. 13A and 13B, matching of the structure shape to the model is not close in FIG. 13A, but FIG. 13B with two trapezoids shows a dramatic increase in the match between the model and the structure shape. There are some further but minor improvements in the models matching the structure shape as the number of trapezoids used increases to three and four.

FIG. 13E illustrates exemplary graphs of the cost function and GOF of simulated diffraction signals versus the measured signals as a function of the number of geometric shapes used in the profile model. Graph 1000 illustrates how the cost function and GOF varies as the number of trapezoids used in the profile model is increased. As can be seen in the cost function graph 1004, the cost function value of modeling the structure depicted in FIG. 13A with one trapezoid is relatively high at 3.0. The cost function graph 1004, using the left Y-axis, drops dramatically to about 1.5 with two trapezoids, less as the number of trapezoids increases from two to three and from three to four trapezoids. The GOF graph 1002, using the right Y-axis, increases dramatically from a GOF of about 0.920 to 0.97 when the number trapezoids increases from one to two, less as the as the number of trapezoids increases from two to three and from three to four trapezoids. As discussed previously, the profile model selection determines the simplest combination of geometric shapes in the profile model that meets or exceeds the termination criteria, which may be a cost function value and/or a GOF value. As also discussed above, the profile model may be a combination of different types of geometric shapes, where trapezoid is just one possible shape that can be used.

FIG. 14 is an exemplary graph of the cost function and GOF of simulated diffraction signals versus the measured signals as a function of the number of parameters used in the profile model. Graph 1100 illustrates how the cost function and GOF varies as the number of parameters used in the profile model is increased. As can be seen in the cost function graph 1104, the cost of modeling a hypothetical structure with three parameters is relatively high at 2.9. The cost function graph 1104, using the left Y-axis, drops dramatically to about 1.6 with five parameters, less as the number of parameters increases from five to six and from six to seven parameters. The GOF graph 1102, using the right Y-axis, increases dramatically from a GOF of about 0.915 to 0.965 when the number of parameters increased from three to five, less as the number of parameters increases from five to six and from six to seven parameters.

As discussed previously, the optimization parameter selection selects parameters that are uncorrelated, have high sensitivity, and allows detection of the change in parameter size required by the application. The selected optimization parameters of the profile model are used to simulate the diffraction signals for different profile dimensions, and the simulated diffraction signals are compared to the corresponding measured signals to calculate the cost function and GOF. Once the profile model selected and the selected optimization parameters of the selected profile model provide simulated diffraction signals results that meet or exceed the termination criteria, then the selection process is complete. As discussed above, the regression results such as CD's, film thickness, and profile from the parameter selector 1830 of FIG. 6A may be used by a system user to fine-tune the recipe or fabrication process. Alternatively, the regression results may be used to adjust variables and/or physical controls of the fabrication process. As also noted above, the profile model and optimization parameters selected may be used to create a library of simulated signals and associated profile data.

FIG. 15 is a storage layout of data store layout in an exemplary embodiment. The data store format 1200 for selected model and parameters includes fabrication process, wafer site, structure, and optical metrology device identification data 1210. The data store format 1200 may include one or more data segments, each data segment comprising the termination criteria 1220, selected model identification 1230 and optimization parameter selection criteria 1240, and selected optimization parameters 1, 2, . . . n, 1250. For example, the model identification may be Shallow Trench Isolation Single Trapezoid Model, termination criteria may include a cost function of 1.5 and GOF of 0.995, optimization parameter selection criteria may be a correlation coefficient of 0.50 and sensitivity of 0.01 SSE, and the selected optimization parameters may be resist top CD, resist bottom CD, resist thickness, anti-reflective coating thickness, and silicon dioxide thickness.

It is contemplated that functional implementation of the exemplary embodiments described herein may be implemented equivalently in hardware, software, firmware, and/or other available functional components or building blocks. Other variations and embodiments are possible in light of above teachings, and it is thus intended that the scope of invention not be limited by this Detailed Description, but rather by Claims following. 

1. A method of selecting a profile model for use in optical metrology of a structure formed on a wafer, the method comprising: obtaining a profile model having a set of geometric parameters associated with dimensions of the structure; converting the set of geometric parameters to a set of optimization parameters, wherein the number of optimization parameters within the set of optimization parameters is less than the number of geometric parameters within the set of geometric parameters; selecting a set of selected optimization parameters from the set of optimization parameters, wherein the optimization parameters of the set of selected optimization parameters are used as parameters of the profile model; and testing the profile model, which is defined using the optimization parameters of the set of selected optimization parameters, against one or more termination criteria.
 2. The method of claim 1, wherein if the one or more termination criteria are not met, selecting another set of selected optimization parameters.
 3. The method of claim 1, wherein selecting a set of selected optimization parameters includes: determining correlation between optimization parameters in the set of optimization parameters.
 4. The method of claim 1, wherein selecting a set of selected optimization parameters includes: determining a confidence interval for each optimization parameter in the set of optimization parameters.
 5. The method of claim 1, wherein selecting a set of selected optimization parameters includes: determining sensitivity of simulated diffraction to changes in one or more optimization parameters in the set of optimization parameters.
 6. The method of claim 1, wherein selecting a set of selected optimization parameters includes: determining correlation between optimization parameters in the set of optimization parameters; determining a confidence interval for each optimization parameter in the set of optimization parameters; and determining sensitivity of simulated diffraction to changes in one or more optimization parameters in the set of optimization parameters.
 7. The method of claim 1, wherein testing the profile model includes: testing if a simulated diffraction signal cost function value is less than or equal to a preset cost function value, wherein the simulated diffraction cost function value is calculated by comparing an optimized simulated diffraction signal to a measured diffraction signal, wherein the optimized simulated diffraction signal is generated using the profile model.
 8. The method of claim 1, wherein testing the profile model includes: testing if a simulated diffraction signal goodness of fit value is less than or equal to a preset goodness of fit value, wherein the simulated diffraction goodness of fit value is calculated by comparing an optimized simulated diffraction signal to a measured diffraction signal, wherein the optimized simulated diffraction signal is generated using the profile model.
 9. The method of claim 1, wherein testing the profile model includes: testing if one or more calculated confidence interval values are less than or equal to corresponding preset confidence interval values, the confidence interval being a range of values of an optimization parameter within which an actual value is expected to fall within a specified probability.
 10. The method of claim 1, wherein testing the profile model includes: testing if a simulated diffraction signal cost function value is less than or equal to a preset cost function value, wherein the simulated diffraction cost function value is calculated by comparing an optimized simulated diffraction signal to a measured diffraction signal, wherein the optimized simulated diffraction signal is generated using the profile model; testing if a simulated diffraction signal goodness of fit value is less than or equal to a preset goodness of fit value, wherein the simulated diffraction goodness of fit value is calculated by comparing an optimized simulated diffraction signal to a measured diffraction signal, wherein the optimized simulated diffraction signal is generated using the profile model; and testing if one or more calculated confidence interval values are less than or equal to corresponding preset confidence interval values, the confidence interval being a range of values of an optimization parameter within which an actual value is expected to fall within a specified probability.
 11. A computer-readable storage medium containing computer executable instructions for selecting a profile model for use in optical metrology of a structure formed on a wafer, comprising instructions for: obtaining a profile model having a set of geometric parameters associated with dimensions of the structure; converting the set of geometric parameters to a set of optimization parameters, wherein the number of optimization parameters within the set of optimization parameters is less than the number of geometric parameters within the set of geometric parameters; selecting a set of selected optimization parameters from the set of optimization parameters, wherein the optimization parameters of the set of selected optimization parameters are used as parameters of the profile model; and testing the profile model, which is defined using the optimization parameters of the set of selected optimization parameters, against one or more termination criteria.
 12. The computer-readable storage medium of claim 11, wherein if the one or more termination criteria are not met, selecting another set of selected optimization parameters.
 13. The computer-readable storage medium of claim 11, wherein selecting a set of selected optimization parameters includes instructions for: determining correlation between optimization parameters in the set of optimization parameters.
 14. The computer-readable storage medium of claim 11, wherein selecting a set of selected optimization parameters includes instructions for: determining a confidence interval for each optimization parameter in the set of optimization parameters.
 15. The computer-readable storage medium of claim 11, wherein selecting a set of selected optimization parameters includes instructions for: determining sensitivity of simulated diffraction to changes in one or more optimization parameters in the set of optimization parameters.
 16. The computer-readable storage medium of claim 11, wherein testing the profile model includes instructions for: testing if a simulated diffraction signal cost function value is less than or equal to a preset cost function value, wherein the simulated diffraction cost function value is calculated by comparing an optimized simulated diffraction signal to a measured diffraction signal, wherein the optimized simulated diffraction signal is generated using the profile model.
 17. The computer-readable storage medium of claim 11, wherein testing the profile model includes instructions for: testing if a simulated diffraction signal goodness of fit value is less than or equal to a preset goodness of fit value, wherein the simulated diffraction goodness of fit value is calculated by comparing an optimized simulated diffraction signal to a measured diffraction signal, wherein the optimized simulated diffraction signal is generated using the profile model.
 18. The computer-readable storage medium of claim 11, wherein testing the profile model includes instructions for: testing if one or more calculated confidence interval values are less than or equal to corresponding preset confidence interval values, the confidence interval being a range of values of an optimization parameter within which an actual value is expected to fall within a specified probability.
 19. A system for selecting a profile model for use in optical metrology of a structure formed on a wafer, comprising: a model generator configured to generate a profile model having a set of geometric parameters associated with dimensions of the structure, and convert the set of geometric parameters to a set of optimization parameters is less than the number of geometric parameters within the set of geometric parameters; a parameter selector configured to select a set of selected optimization parameters from the set of optimization parameters, wherein the optimization parameters of the set of selected optimization parameters are used as parameters of the profile model; and a profile model tester configured to test the profile model, which is defined using the optimization parameters of the set of selected optimization parameters, against one or more termination criteria. 